Ersatz gravity and black-hole thermodynamics from Manin gauge theory with noncompact gauge group

We show that a three-dimensional Manin gauge theory with gauge group $\operatorname{SL}(2;\mathbb R)$ (i.e. Yang-Mills theory, the third-way theory, or the imaginary third-way theory) minimally coupled to Einstein gravity admits a dual interpretation as Einstein gravity with an exotic coupling to a Manin gauge theory, where the roles of dreibein/spin connection and field strength/gauge potential are interchanged. The dual, or ersatz, gravitational metric $\hat g_{\mu\nu}\sim\operatorname{tr}((\star F)_\mu (\star F)_\nu)$ is a classical double copy of the gauge field strength $F_{\mu\nu}$ (as opposed to the usual double copy of the gauge potential $A_\mu$). If matter exclusively couples to $\hat g$ (for example, in a gravitational decoupling limit), then one can formulate black-hole thermodynamics with regards to the ersatz metric. In particular, a black-hole solution for the ersatz metric $\hat g_{\mu\nu}$ (made of Yang-Mills fields) radiates ersatz Hawking radiation and obeys the laws of black-hole thermodynamics.